Kerr-Newman scalar clouds

Massive complex scalar fields can form bound states around Kerr black holes. These bound states—dubbed scalar clouds—are generically nonzero and finite on and outside the horizon; they decay exponentially at spatial infinity, have a real frequency and are specified by a set of integer “quantum” numbers $(n,l,m)$. For a specific set of these numbers, the clouds are only possible along a one-dimensional subset of the two-dimensional parameter space of Kerr black holes, called an existence line. In this paper we make a thorough investigation of the scalar clouds due to neutral (charged) scalar fields around Kerr(-Newman) black holes. We present the location of the existence lines for a variety of quantum numbers, their spatial representation and compare analytic approximation formulas in the literature with our exact numerical results, exhibiting a sometimes remarkable agreement.

Figure 1

Existence lines for scalar clouds with various quantum numbers in the mass vs horizon angular velocity parameter space of Kerr BHs. The black solid curve represents the extreme case, $a=M$ and Kerr solutions exist below this line. The blue dashed and red dotted lines represent the nodeless solution for $m=l=1$ and $m=l=2$, respectively. The insets compare the nodeless solutions ($n=0$) with the solutions with $n=1,2$.

Figure 2

Analogous plot to Fig. 1, but now the insets compare the solutions for $m=l$ with the solutions with $m<l$, all with $n=0$.

Figure 3

Comparison between our numerical solution for the clouds with $n=0$, $m,l=1$ and the analytical results by Hod [11], cf. Eq. (14), and Detweiler [3], cf. Eq. (16), in a mass vs angular momentum parameter for Kerr BHs.

Figure 4

Analogous comparison to that in Fig. 3 but now for the clouds with $n=0$, $m=l=3,4$. The agreement between the analytic and numerical approximations seems to become slightly worse, outside their regime of validity, when the quantum numbers $m=l$ increase.

Figure 5

Existence lines for charged scalar bound states in the Kerr-Newman background, for $n=0$ and $l=m=1$, for different values of the field charge and fixed background charge $\mu Q=0.1$.

Figure 6

Existence lines for charged scalar bound states in the Kerr-Newman background, for $n=0$ and $l=m=1$, for different values of the background charge and fixed field charge $q/\mu =1$.

Figure 7

Existence lines for charged scalar bound states in the Kerr-Newman background, for $\mu Q=0.1$, $q/\mu =1$, $n=0$ and $m=l=1,2$ and 3.

Figure 8

Comparison between our numerical solutions and the analytical formula by Furuhashi and Nambu [27], cf. Eq. (18), for clouds with $n=0$, $m=l=1$ and $q{r}_{+}=0.1$. The numerical and analytical formulas coincide with very good accuracy.

Figure 9

Three-dimensional plots of the spheroidal harmonics $|S_{lm}(\theta )|$. All three panels were obtained for $\mu r_{+}=0.5$. These correspond, from left to right to $\mu a=0.399$, $\mu a=0.133$ and $\mu a=0.430$.

Figure 10

Radial solutions $R_{11}$ (left), $R_{22}$ (right) for clouds with $n=0,1,2$ in the Kerr background with $\mu r_{+}=0.5$. The corresponding values of $\mu a$ are given in the figure key.

Figure 11

Radial solutions $R_{11}$ (left), $R_{22}$ (right) for clouds with $q/\mu =1$ and $n=0,1,2$ in the background of Kerr-Newman with $\mu r_{+}=0.5$, $Q\mu =0.1$. The corresponding values of $\mu a$ are given in the figure key.

Figure 12

Position of the clouds, $r_{MAX}/M$ (see definition in the text), and of the corotating CNG, as a function of $a/M$ for clouds with $n=0$ and $l=m=1,2,3,4,5,10$ in the Kerr background.

Figure 13

Three-dimensional spatial distribution of a cloud (left panel) with $n=0$ and $m=l=1$ and its energy density (right panel) in terms of polar coordinates $(\rho ,z)$. Both are essentially supported along an equatorial torus, due to the angular distribution of the corresponding spheroidal harmonic.